Normalized defining polynomial
\( x^{10} - 2 x^{9} + 523 x^{8} - 834 x^{7} + 111104 x^{6} - 132956 x^{5} + 11978835 x^{4} + \cdots + 14552228339 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-2937451902360726175744\) \(\medspace = -\,2^{15}\cdot 11^{8}\cdot 53^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(140.22\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{3/2}11^{4/5}53^{1/2}\approx 140.21583946452205$ | ||
Ramified primes: | \(2\), \(11\), \(53\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-106}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(4664=2^{3}\cdot 11\cdot 53\) | ||
Dirichlet character group: | $\lbrace$$\chi_{4664}(1,·)$, $\chi_{4664}(2755,·)$, $\chi_{4664}(1907,·)$, $\chi_{4664}(1697,·)$, $\chi_{4664}(2121,·)$, $\chi_{4664}(1483,·)$, $\chi_{4664}(3393,·)$, $\chi_{4664}(2545,·)$, $\chi_{4664}(1059,·)$, $\chi_{4664}(4027,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-106}) \), 10.0.2937451902360726175744.1$^{15}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{20\!\cdots\!73}a^{9}+\frac{83\!\cdots\!89}{20\!\cdots\!73}a^{8}-\frac{32\!\cdots\!57}{20\!\cdots\!73}a^{7}+\frac{27\!\cdots\!76}{20\!\cdots\!73}a^{6}+\frac{80\!\cdots\!73}{20\!\cdots\!73}a^{5}-\frac{72\!\cdots\!35}{20\!\cdots\!73}a^{4}+\frac{28\!\cdots\!79}{20\!\cdots\!73}a^{3}+\frac{84\!\cdots\!08}{20\!\cdots\!73}a^{2}-\frac{82\!\cdots\!57}{20\!\cdots\!73}a-\frac{90\!\cdots\!78}{20\!\cdots\!73}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{148830}$, which has order $148830$ (assuming GRH)
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{42\!\cdots\!70}{20\!\cdots\!73}a^{9}-\frac{76\!\cdots\!93}{20\!\cdots\!73}a^{8}+\frac{22\!\cdots\!14}{20\!\cdots\!73}a^{7}-\frac{31\!\cdots\!59}{20\!\cdots\!73}a^{6}+\frac{50\!\cdots\!20}{20\!\cdots\!73}a^{5}-\frac{50\!\cdots\!45}{20\!\cdots\!73}a^{4}+\frac{61\!\cdots\!46}{20\!\cdots\!73}a^{3}-\frac{36\!\cdots\!74}{20\!\cdots\!73}a^{2}+\frac{29\!\cdots\!88}{20\!\cdots\!73}a-\frac{10\!\cdots\!93}{20\!\cdots\!73}$, $\frac{51\!\cdots\!00}{20\!\cdots\!73}a^{9}-\frac{11\!\cdots\!48}{20\!\cdots\!73}a^{8}+\frac{43\!\cdots\!44}{20\!\cdots\!73}a^{7}-\frac{47\!\cdots\!89}{20\!\cdots\!73}a^{6}+\frac{10\!\cdots\!22}{20\!\cdots\!73}a^{5}-\frac{74\!\cdots\!80}{20\!\cdots\!73}a^{4}+\frac{97\!\cdots\!86}{20\!\cdots\!73}a^{3}-\frac{53\!\cdots\!09}{20\!\cdots\!73}a^{2}+\frac{33\!\cdots\!46}{20\!\cdots\!73}a-\frac{14\!\cdots\!48}{20\!\cdots\!73}$, $\frac{17\!\cdots\!94}{20\!\cdots\!73}a^{9}+\frac{12\!\cdots\!08}{20\!\cdots\!73}a^{8}+\frac{70\!\cdots\!42}{20\!\cdots\!73}a^{7}+\frac{76\!\cdots\!80}{20\!\cdots\!73}a^{6}+\frac{11\!\cdots\!50}{20\!\cdots\!73}a^{5}+\frac{20\!\cdots\!92}{20\!\cdots\!73}a^{4}+\frac{80\!\cdots\!12}{20\!\cdots\!73}a^{3}+\frac{23\!\cdots\!29}{20\!\cdots\!73}a^{2}+\frac{21\!\cdots\!14}{20\!\cdots\!73}a+\frac{10\!\cdots\!24}{20\!\cdots\!73}$, $\frac{51\!\cdots\!00}{20\!\cdots\!73}a^{9}-\frac{11\!\cdots\!48}{20\!\cdots\!73}a^{8}+\frac{43\!\cdots\!44}{20\!\cdots\!73}a^{7}-\frac{47\!\cdots\!89}{20\!\cdots\!73}a^{6}+\frac{10\!\cdots\!22}{20\!\cdots\!73}a^{5}-\frac{74\!\cdots\!80}{20\!\cdots\!73}a^{4}+\frac{97\!\cdots\!86}{20\!\cdots\!73}a^{3}-\frac{53\!\cdots\!09}{20\!\cdots\!73}a^{2}+\frac{33\!\cdots\!46}{20\!\cdots\!73}a-\frac{14\!\cdots\!75}{20\!\cdots\!73}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 26.1711060094 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 26.1711060094 \cdot 148830}{2\cdot\sqrt{2937451902360726175744}}\cr\approx \mathstrut & 0.351881571118 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 10 |
The 10 conjugacy class representatives for $C_{10}$ |
Character table for $C_{10}$ |
Intermediate fields
\(\Q(\sqrt{-106}) \), \(\Q(\zeta_{11})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }$ | R | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.1.0.1}{1} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.1.0.1}{1} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }$ | R | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.15.13 | $x^{10} - 12 x^{9} + 174 x^{8} - 640 x^{7} - 40 x^{6} + 11424 x^{5} - 7984 x^{4} - 79360 x^{3} + 84112 x^{2} + 955968 x + 1404384$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
\(11\) | 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
\(53\) | 53.10.5.2 | $x^{10} + 23671443 x^{2} - 21327970143$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |